Ph.D. supervision: Projects
Last update: May 31, 2006
Here is a general description of topics which / I have researched / I
am currently researching / I will eventually be researching / and which
can be of interest for students considering to complete a Ph.D. degree
under my supervision in algebraic topology and related areas. If you are
interested and would like to know more details on these projects, you are
welcome to email me at jesus@math.cinvestav.mx. You can
get an idea of my completed research activities by taking a look at the
list of my publications and to the summarized description of my mathematical
work.
Stable Homotopy Types
Lens Spaces
Stunted real projective spaces can be interpreted as Thom complexes of
multiples of the Hopf bundle. Due to the finiteness of Adams' J-groups,
the stable homotopy types of these spaces are periodic. The stable
classification of these spaces was began in 1977 by Feder-Gitler-Mahowald,
and was completed in 1986 by Davis-Mahowald by using extensive
calculations with the classical Adams spectral sequence. In my PhD thesis
(1994) I showed how to get, in a more natural way, the same result using
instead the Adams spectral sequence based at connective (real) K-theory.
Moreover, this approach applied to any prime and, in particular, I
completed the corresponding classification for stunted lens spaces of
p torsion (and their stable summands), with p an odd
prime. In 1998 Yang obtained the corresponding classification for
4-torsion lens spaces by extending the ideas of Davis and Mahowald. The
complete stable classification in the general case (say powers of any
prime) is still an open problem well suited for a doctoral work. I this
direction, I obtained in 2000 partial results for odd primes by mixing
homotopical and "combinatorial" considerations (these results applied in
fact to both lens spaces and p-localizations of complex
projective spaces). Yet, I think much more results should be obtainable by
combining the classical Adams spectral sequence calculations with the
corresponding calculations based at the spectrum for connective K-theory.
Connective K-Theoretic Adams Spectral Sequence
A key step in the above program is the complete determination of the
differentials holding in the Adams spectral sequence based at connective
K-theory for lens spaces. It should be possible to gather this much of the
information from the current literature. A second and more serious step is
to extend, from any prime to arbitray prime powers, the calculations that
describe the "vanishing line" that holds in the spectral sequence --this
much of the program is interesting in its own right. Such a calculation
should shed considerable light into the description of the part of the
stable homotopy groups of spheres relevant for the classification of the
stable homotopy types of general stunted lens spaces. In this same
direction, it is to be observed that the calculations using this vanishing
line work for primes larger than 7. For instance, at the prime 2, one
needs to improve the slope of the line --this was accomplished in 1981 by
Mahowald. Thus, another interesting point here is to analyze the
corresponding situation for the cases p=3,5,7.
Euclidean Immersions of Certain Manifolds
A classical problem in differential topology is to determine the dimension
of the smallest
Euclidean space where a given manifold admits an immersion. A natural
family of manifolds where to try this problem is formed by the (real,
complex, quaternionic) projective spaces. Since 1960, with the work of
José Adem and Samuel Gitler, my institution has been one of the
leader research centers in this area. The general solution to this
problem is extremely complicated and still open and, consequently, it is
an obvious place where to test the power of new techniques in mathematics.
Much of my recent work has to do with the immersion problem for lens
spaces, as they sit half way between the real version of projective
spaces (whose immersion problem is amazingly complicated) and the
corresponding complex
version (whose immersion problem, although still open, seems to be more
accesible). There have been a number of approaches to this immersion
problems, and many of them continue giving new results. Here is a general
list of methods I have been using:
Classical Obstruction Theory
Hirsch's theorem claims that the immersion question for a manifold is a
problem in homotopy theory: it's just a matter of determining the
geometric dimension of the stable normal bundle of the manifold, that is,
lifting the corresponding classifying map from BO to a space BO(k)
--smallest possible such k. Obstruction theory is a classical
cohomological tool in
homotopy theory for lifting maps and, thus, it is natural to approach the
immersion problem in this way. Between 1964 and 1966 Gitler and Mahowald
developed a modified version of obstruction theory (modified Postnikov
towers) which is better adapted to the explicit computations required
in the immersion problem. This approach has lead to many substancial
results by a number of people and, in particular, I have obtained new
results in the cases of
lens spaces. The method is under continuous use and gives interesting
challenges for a doctoral research.
Axial Maps and Topological Complexity
There are many equivalent formulations for the immersion problem for real
projective spaces, one of which (due to Adem-Gitler-James) is
particularly attractive dues to the simplicity of its statement. It
involves the existence of suitable axial maps --compressions of the Hopf
multiplication in (skeleta of) the Eilenberg-MacLane space of type
(Z/2,1). In recent work with Astey and Davis, I have developped the
correct notion of axial maps of higher 2-torsion, giving an alternative
statement for the immersion problem for lens spaces. Thus, a possible PhD
topic is the construction of immersions by constructing explicit
(generalized) axial maps. There are two ways in which this can be
performed: again using obstruction theory (in the form of sectioning
certain bundles), or by means of linear methods --it is interesting to
remark that the strongest general immersions constructed for
projective spaces where obtained in 1967 by Milgram extending the ideas of
bilinear maps (linear versions of axial maps). The work on axial maps
turns out to be
closely related to the Topological Complexity of projective and lens
spaces, a concept arising naturally in the study of motion
planning (robotics) in the space of configurations of
the roots of unity in a complex Euclidean space. Together with L.
Zárate, a Ph.D. student working under my suervision, we have
recently used
this language as an alternative approach to the immersion problem for lens
spaces in general, and proyective spaces in particular. Yet, this
direction of research is full of new options to try, specially those
related to bordism matters (described below).
Secondary Operations in K-Theory
Although classical obstruction theory was originally designed to work in
singular cohomology, the methods can be adapted to any suitable
generalized cohomology theory. In 1977 Feder and Iberkleid bagan such an
approach based in complex K-theory in order to study the geometrical
dimension of multiples of the real Hopf bundle, an issue closely related
to the immersion problem for projective spaces. This technique allowed
them to reprove a number of the Ph.D. Davis' homotopical results with a
theoretically and computationally easier approach. Even more,
Feder-Iberkleid's K-theory-secondary-operations ideas give, on the nose, a
natural way to study the generalized vector field problem for lens spaces.
Such an approach is particularly interesting since much information is
expected to be drawn on the role of the 2-torsion within the immersion
problem for lens spaces. This represents joint work in progress with M.
Velasco-Fuentes, a Ph.D. student working under my spervision.
Bordism Euler Classes
One further step in the direction of generalized obstruction theory can be
done in terms of bordism of manifolds. In 1984 Don Davis showed how to
effectively use a prime-localized version of bordism theory, namely
Brown-Peterson theory (which will be discussed a bit more below), in order
to study the immersion problem for projective spaces. Davis' result gives,
in an amazingly simple statement, what is by now the best general
nonimmersion results for these manifolds. The idea is strikingly simple
and one of its interpretations can be easily described as follows: show
the non-triviality of the Brown-Peterson Euler class of a certain vector
bundle which should have a nowhere trivial section whenever a certain
projective space admits an Eucliden immersion. In 2003 I studied the
analogous problem for higher 2-torsion lens spaces, extending Davis'
method. This gave a renewed interest to the analysis of the immersion
problem with an eye on the role of the 2-torsion. The philosophy behind
this is to analyze an extremely hard problem (immersions of projective
spaces ---manifolds "built enterely in terms of exact 2-torsion") by
approaching it via more accesible variations (immersions of
lens spaces ---manifolds "built in terms of higher 2-torsion"). The
generalized methods required a rather complete study of the
2-divisibility properties in the formal group law associated to
Brown-Peterson theory (this topic is described in some more detail below).
It should be remarked that the generalization of Davis' results faced
certain technical problems which did not allow me to get the
result in its full expected generality. In any case, completing the
missing details toward the unrestricted theorem would be an
interesting Ph.D. project to work on.
Stiefel Manifolds
As shown by Gitler in 1968, another equivalent formulation for the
immersion problem for real projective spaces is given in terms of the
projective Stiefel manifolds. As a consequence of my work on
generalized axial maps, it follows that there is a corresponding analogue
for lens spaces, in terms of certain "lentified Stiefel manifolds". I have
not explored yet this idea in its full generality, but it should lead to
new results. In particular, the study of these manifolds is interesting in
its own right. For instance, a possible PhD work has to do with studying
the corresponding parallelizability properties of the lentified Stiefel
manifolds. The idea is motivated by recent work of
Astey-Gitler-Micha-Pastor which solves the last question for the
projectivized Stiefel manifolds. As in the case of the immersion problem,
it turns out that the solution is simpler in the complex case than in the
real case. Thus, I expect that the parallelizability of high torsion
lentified Stiefel manifolds should be quite accesible.
Formal Group Laws
The theory of formal group laws has shown to be of special importance in
mathematics mainly due to the wide variety of connections it has had with
other mathematical branches like geometry, algebraic topology, number
theory and combinatorics. One of the basic connections arises through the
universal example: a formal group law F over the Lazard ring
L which contains as much information as any other formal group.
The relation to topology comes from the fundamental theorem of
Milnor-Quillen which claims that L is the homotopy ring of
complex cobordism and that, in these terms, F is the universal
Euler class for complex-oriented cohomology theories. This bridge has led
to a number of basic developments. For instance, Hopkins-Miller have used
a partial converse of Quillen's theorem in constructing higher K-theories
related to (the formal groups associated to certain) elliptic curves
--this will be commented in the next section.
Arithmetics
Much of my recent work has been directed to the determination of a number
of algebriac properties of formal groups and its application to
differential topology. This is performed by analizing, in fact, the
universal formal group by means of its associated series --iterated formal
sums of a single variable. Topologically, these associated series can be
interpreted as the Euler classes of powers of the Hopf bundle and,
therefore, the determination of their algebriac properties is fundamental
in the study of the cobordism of lens spaces. In doing this, it is
convenient to consider one prime at a time, and topologically this
corresponds to splitting cobordism theory into its primary building
blocks: Brown-Peterson homology. The resulting algebraic objets are
powers series with coefficients in a p-local polynomial ring on
an infinite number of variables --a localized version of the Lazard ring.
Following pionering work of D. C. Johnson, in a series of papers I have
determined the p-divisibility of such coefficients, in fact
describing much of the polynomial structure of each coeffient in the
associated series. The
method requires to filter the polynomial
ring and study the coefficients in the (simpler) associated graded object.
Several different filtrations have been used and many others are awaiting
to be analyzed --this is part of the Ph.D. work of L. Zárate under
my supervision.
Further Algebraic Connections and Developments
Number theory is perhaps the area most naturaly linked to the theory of
formal groups, and the relations have become abundant over the time. The
text by Hazewinkel gives an excellent revision for known applications (up
to the mid 70's) of formal group theory into number theory as well as
into arithmetical and algebriac geometry. More recent applications to
cryptography, where it is important to have methods for computing the
cardinality of the group of rational points of elliptic curves defined
over a finite field F, can be derived from the results in the
Ph.D. thesis of Couveignes. In that work, formal group laws associated to
elliptic curves are used to give effective methods to compute isogenies.
More generally, it is possible to associate formal group laws to
algebraic varieties. Most interesting cases seem to be one-dimensional
formal groups arising from Calabi-Yau varieties. For instance, the
p-series (and in particular the pth coefficient)
contains information about the number of rational points on the variety
over the field with p elements. Then, a natural Ph.D. topic would
consist on extending these properties to higher powers of primes. Formal
group theory has also proven to have close connections to class field
theory, offering alternative approaches which reveal remarkable properties
of number and local fields (see the book by Fesenko and Vostokov).
In combinatorics it is worth noticing the interrelation of formal
group theory with umbral calculus (developped by Ray and his coworkers).
As for applications in other areas of mathematics, the theory of formal
groups has played, in fact, a sort of unifying role. Since the 1986
conference at the IAS in Princeton ---whose original aim centered at (that
time) recent developments of elliptic genera and elliptic cohomology--- it
has became clear that geometry and physics enter prominently into the
subject. In particular, algebraic topology has seen deep connections to
those areas via the formal-group-grounds it shares with number theory.
Relation to Topology: Conner Floyd Conjecture and Kervaire
Invariant 1
A deep connection between algebraic and differential topology starts with
the study of bordism classes of free actions on oriented manifolds. This
problem led Conner and Floyd to consider the oriented bordism homology of
B(n,p), the iterated n-fold smash product of the
classifying space for the cyclic group C(p) of order p.
As they noticed, the bottom ``toral'' class in these groups plays a
fundamental role in the problem, for its annihilator ideal I(n)
is generated by those bordism classes of oriented manifolds admitting an
action, free of stationary points, of the n-fold iterated cartesian
product of C(p) with itself ---tha is, the elementary abelian
p-group of rank n. Many of the geometric applications
in Conner and Floyd's theory can be recovered provided a conjectured
description of I(n) holds (the so-called Conner-Floyd
conjecture). For this problem one can replace the Thom spectrum
MSO by the Brown-Peterson spectrum BP --the building
block of the p-localized complex cobordism-- and, in these terms,
the p-series plays a major role, and its relevance has been
confirmed by Minami's work on the possible existence of framed manifolds
of Kervaire invariant 1 (that is, on the basic problem of understanding
stable homotopy classes of spheres detected in the 2-line of the classical
Adams spectral sequence). In view of the basic role the p-series
played in the above development, an interesting PhD project would be to
analyze the extent to which the p^k-series can be used in a
similar calculation as above. As a first alternative of this, in joint
work with my Ph.D. student L. Zárate, we have extended the
p-divisibility properties in the coefficients of the universal p-typical
p^k series in order to describe a close interaction between the
p-divisibility and the v_1-divisibilitiy properties in the
coefficients of that series. The outcome has been a description of what
seems to be the optimal relations in the BP-homology of the classifying
space for a general abelian p-group of rank 2. The final goal here (still
in progress) is to give an explicit description of the annihilator ideal
in the BP-homology of such a group (the classical
Conner-Floyd conjecture described above deals only with
elementary groups --of arbitrary rank, though).
Having described my research up to this point, it is now
interesting to remark that, in fact, the work just described on the
generalized Conner-Floyd conjecture for the group C(2^e)
X C(2^e) was
motivated by the motion planning approach (described
above) to the immersion problem for projective spaces. In retrospect, the
above analysis is meant to substitute the technical dificulties arising in
the computation of the BP-homology of the "non-symmetric" group C(2^e)
X C(2) arising in the axial map approach, through 2-torsion lens
spaces, to the immersion problem for
projective spaces. Indeed, the relevant group in the corresponding
topological complexity approach is precisely C(2^e)
X C(2^e).
Stable Homotopy Theory, Algebraic Geometry and Number
Theory
Topological Modular Forms
Manifolds --oriented, (stably-almost) complex, Spin, ...-- can be
approached through algebraic models called Hirzebruch's genera. These are
ring homomorphisms invariant under cobordism. The most famous examples
are probably the Z/2 orientation of manifolds MSO --> Z/2, the Todd genus
MSO --> Z, and the Atiyah-Bott-Shapiro genus
MSpin --> KO (the last one already made periodic). For instance, the
Atiyah-Singer index theorem, says that the
latter can be calculated analytically by taking the index of the Dirac
operator on the manifold. The relation with stable homotopy theory comes
from the fact that both of these two genera have standard relizations as
maps of spectra.
Around 1985, Ochanine introduced his elliptic genus, which assigns a
level-2 modular form to an oriented manifold. Motivated by the
Atiyah-Singer index theorem, Witten generalized this and produced a genus
from MString (the cobordism ring of Spin manifolds whose halved
first Pontryagin class is trivial) to the ring of q-expansions of
modular forms (mathematical ideas first formalized by Bott-Taubes). In
1995 Ochanine-Witten fundamental work found deep connections with homotopy
theory: Hopkins, Mahowald and Miller have constructed a new family of
spectra wich play a fundamental role in realizing Witten's genus. The
outcome has been a spectrum called tmf (for "topological
modular
forms") which has opened a whole new view in the homotopy word.
Rough details are as follows: As mentioned above, the formal groups
associated to certain elliptic curves produce elliptic spectra analogous
to complex K-theory. Unlike the case in K-theory, there
does not seem to be sufficient reason to prefer one elliptic spectrum over
another. This is related to the fact that there is no moduli space of
elliptic curves, but only a moduli stack. One can then imitate the moduli
stack idea and construct tmf as the homotopy inverse limit of all
the elliptic spectra --making these ideas precise requires a great deal of
tools and work.
Much of my current research has to do with the construction,
properties and applications of tmf --this spectrum has already
shown a remarkable importance in the computation of the stable homotopy
groups of spheres. For instance, in a recent paper, Bruner-Davis-Mahowald
have obtained a strong especialization of Davis' nonimmersion result for
projective spaces (discussed above). Thus, a possible PhD topic would be
the extension of Bruner-Davis-Mahowald techniques to the immersion problem
for lens spaces. In this respect, another very interesting project here
would be the complete computation of the tmf cohomology of
products of real projective spaces (and lens spaces). This would possibly
require considerable use of Adams-Novikov type spectral sequences and, as
a bonus, the development of a non complex-oriented analogue of the
universal 2-series.
Higher Chromatic Phenomena
Much of the success in the theory described above comes from the rich
structure of algebraic curves. However, to make the theory work, one has
to restrict attention to elliptic curves. In the homotopy word,
this means restricting attention to the second level in the chromatic
approach to the stable homotopy groups of spheres (v_2 periodic phenomena)
. In a recent paper, Ravenel addresses the question of "attaching formal
groups of height grater than 2 to algebraic curves (of genus grater than
2) to get insights into the cohomology theories that go deeper into the
chromatic tower". Part of my research is now following this path.